Is there a way to play back the recorded phone call directly from the phone? For example, if I go into the History, can I add an option to retrieve the recorded call and play it back through the phone?
No. But it could theoretically be done. Open a feature request.
Oooooh I really like that idea. Open a feature request at issues.freepbx.org under the phone project. That would be a great additional feature we could build into the Sangoma Phones when paired with FreePBX.